The phenomenological theory of sedimentation describes a flocculated suspension as a mixture of the solid and the fluid as two superimposed continuous media. Starting from the mass and linear momentum balances for each component, this theory yields, through constitutive assumptions and an order-of-magnitude analysis, three coupled partial differential equations describing the sedimentation-consolidation behaviour of the suspension in several space dimensions. The study and numerical solution of this system of equations has started only recently, but results are available for the one-dimensional case, in which these modelling equations reduce to a scalar hyperbolic-parabolic strongly degenerate partial differential equation with appropriate initial and boundary conditions. In this contribution, the research work performed by several groups of mathematicians on the formulation, analysis and numerical solution of mathematical sedimentation models is reviewed, with an emphasis on theoretical and numerical results for the simulation of the behaviour of compressible slurries.